arXiv.orghttp://arxiv.org/icons/sfx.gifhttp://arxiv.org/
Solving Random Systems of Quadratic Equations with Tanh Wirtinger Flow. (arXiv:1905.09320v1 [math.OC])http://arxiv.org/abs/1905.09320
<p>Solving quadratic systems of equations in n variables and m measurements of
the form $y_i = |a^T_i x|^2$ , $i = 1, ..., m$ and $x \in R^n$ , which is also
known as phase retrieval, is a hard nonconvex problem. In the case of standard
Gaussian measurement vectors, the wirtinger flow algorithm Chen and Candes
(2015) is an efficient solution. In this paper, we proposed a new form of
wirtinger flow and a new spectral initialization method based on this new
algorithm. We proved that the new wirtinger flow and initialization method
achieve linear sample and computational complexities. We further extended the
new phasing algorithm by combining it with other existing methods. Finally, we
demonstrated the effectiveness of our new method in the low data to parameter
ratio settings where the number of measurements which is less than
information-theoretic limit, namely, $m &lt; 2n$, via numerical tests. For
instance, our method can solve the quadratic systems of equations with gaussian
measurement vector with probability $\ge 97\%$ when $m/n = 1.7$ and $n = 1000$,
and with probability $\approx 60\%$ when $m/n = 1.5$ and $n = 1000$.
</p>
<a href="http://arxiv.org/find/math/1/au:+Luo_Z/0/1/0/all/0/1">Zhenwei Luo</a>, <a href="http://arxiv.org/find/math/1/au:+Zhang_Y/0/1/0/all/0/1">Ye Zhang</a>Optimum Low-Complexity Decoder for Spatial Modulation. (arXiv:1905.09401v1 [cs.IT])http://arxiv.org/abs/1905.09401
<p>In this paper, a novel low-complexity detection algorithm for spatial
modulation (SM), referred to as the minimum-distance of maximum-length (m-M)
algorithm, is proposed and analyzed. The proposed m-M algorithm is a smart
searching method that is applied for the SM tree-search decoders. The behavior
of the m-M algorithm is studied for three different scenarios: i) perfect
channel state information at the receiver side (CSIR), ii) imperfect CSIR of a
fixed channel estimation error variance, and iii) imperfect CSIR of a variable
channel estimation error variance. Moreover, the complexity of the m-M
algorithm is considered as a random variable, which is carefully analyzed for
all scenarios, using probabilistic tools. Based on a combination of the sphere
decoder (SD) and ordering concepts, the m-M algorithm guarantees to find the
maximum-likelihood (ML) solution with a significant reduction in the decoding
complexity compared to SM-ML and existing SM-SD algorithms; it can reduce the
complexity up to 94% and 85% in the perfect CSIR and the worst scenario of
imperfect CSIR, respectively, compared to the SM-ML decoder. Monte Carlo
simulation results are provided to support our findings as well as the derived
analytical complexity reduction expressions.
</p>
<a href="http://arxiv.org/find/cs/1/au:+Al_Nahhal_I/0/1/0/all/0/1">Ibrahim Al-Nahhal</a>, <a href="http://arxiv.org/find/cs/1/au:+Basar_E/0/1/0/all/0/1">Ertugrul Basar</a>, <a href="http://arxiv.org/find/cs/1/au:+Dobre_O/0/1/0/all/0/1">Octavia A. Dobre</a>, <a href="http://arxiv.org/find/cs/1/au:+Ikki_S/0/1/0/all/0/1">Salama Ikki</a>Learning With Errors and Extrapolated Dihedral Cosets. (arXiv:1710.08223v2 [cs.CR] UPDATED)http://arxiv.org/abs/1710.08223
<p>The hardness of the learning with errors (LWE) problem is one of the most
fruitful resources of modern cryptography. In particular, it is one of the most
prominent candidates for secure post-quantum cryptography. Understanding its
quantum complexity is therefore an important goal. We show that under quantum
polynomial time reductions, LWE is equivalent to a relaxed version of the
dihedral coset problem (DCP), which we call extrapolated DCP (eDCP). The extent
of extrapolation varies with the LWE noise rate. By considering different
extents of extrapolation, our result generalizes Regev's famous proof that if
DCP is in BQP (quantum poly-time) then so is LWE (FOCS'02). We also discuss a
connection between eDCP and Childs and Van Dam's algorithm for generalized
hidden shift problems (SODA'07). Our result implies that a BQP solution for LWE
might not require the full power of solving DCP, but rather only a solution for
its relaxed version, eDCP, which could be easier.
</p>
<a href="http://arxiv.org/find/cs/1/au:+Brakerski_Z/0/1/0/all/0/1">Zvika Brakerski</a>, <a href="http://arxiv.org/find/cs/1/au:+Kirshanova_E/0/1/0/all/0/1">Elena Kirshanova</a>, <a href="http://arxiv.org/find/cs/1/au:+Stehle_D/0/1/0/all/0/1">Damien Stehl&#xe9;</a>, <a href="http://arxiv.org/find/cs/1/au:+Wen_W/0/1/0/all/0/1">Weiqiang Wen</a>Semantic programming: method of $\Delta_0^p$-enrichments and polynomial analogue of the Gandy fixed point theorem. (arXiv:1903.08109v3 [cs.CC] UPDATED)http://arxiv.org/abs/1903.08109
<p>Computer programs fast entered in our life and the questions associated with
the execution of these programs have become the most relevant in our days.
Programs should work efficiently, i.e. work as quickly as possible and spend as
little resources as possible. Most often, such a "measure of efficiency" is the
polynomial program execution time of the length of the input data. Such
programs have great importance in the direction of smart contracts on
blockchain.
</p>
<p>In this article will be introduced the method of $\Delta_0^p$-enrichments
which will show how to switch from the usual polynomial model of
$\mathfrak{M}^{(0)}$ using $\Delta_0^p$-enrichments to a model with new
properties and new elements so that the new model will also be polynomial.
</p>
<p>$\Delta_0^p-$enrichments: $\mathfrak{M}^{(0)}\to ... \to \mathfrak{M}^{(i)}
\to ...\to\mathfrak{M}$
</p>
<p>This method based on theory of semantic programming entered in 1970s and
1980s, academics Ershov and Goncharov and professor Sviridenko.
</p>
<p>New element $w$ for $M^{(i+1)}$ and not in $M^{(i)}$ generate with some
$\Delta_0^p-$formula $\Phi_k$ from family $F_j$ for one place predicate $P_j$:
$\mathfrak{M}^{(i)}\models\Phi_k(w_1,...,w_{n_k})$, where $w$ is finite list $w
= &lt;w_1,...,w_{n_k}&gt;$ and now $\mathfrak{M}^{(i+1)}\models P_j(w)$.
</p>
<p>Then we will create an operator
$\Gamma_{F_{P_1^+},...,F_{P_N^+}}^\mathfrak{M^{(i)}}$ and prove polynomial
analogue of the Gandy fixed point theorem. It allows us to take a different
look on polynomial computability.
</p>
<p>Let $\Gamma^*$: $\Gamma_{F_{P_1^+},...,F_{P_N^+}}^\mathfrak{M}(\Gamma^*) =
\Gamma^*$
</p>
<p>Theorem (polynomial analogue of the Gandy fixed point theorem)
</p>
<p>Fixed point $\Gamma^*$ is $\Delta_0^p-$set and $P_1,...,P_N$ -
$\Delta_0^p-$predicates on $\mathfrak{M}$
</p>
<a href="http://arxiv.org/find/cs/1/au:+Nechesov_A/0/1/0/all/0/1">Andrey Nechesov</a>